I have to admit that I did read a couple Bourbaki books. Or should I say “tried to read” because their texts are unreadable to put it mildly. They kill mathematics by removing the beauty of it and instead provide dry and boring texts that could be read by robots. Please don’t try to understand mathematics by reading Bourbaki texts. This is not what mathematics is about.
Unfortunately Bourbakism polluted teaching of Mathematics in Europe c.a. 1960s. With visible effects. Except for Eastern Europe were it was opposed by Arnold.
> These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology.
> The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony of the Universe.
> The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system.
"By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems." (Arnold - On teaching mathematics)
A english translation of this book is available here
I hiked with Arnold once, in New Hampshire. He at one point described walking far enough out onto a frozen lake to watch the ice crack around him, to see if the pattern matched his intuition. He was very hands-on.
> For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system.
This one resonated with me. At one point I had a problem that had to do with Kahler manifolds, but I knew nothing about them and only had some worked examples using basic ideas from Hamiltonian mechanics that I learned from Arnol'd's book. In hopes of resolving my issues, I spoke to some symplectic geometers and used Hamilton-Jacobi language. They vaguely knew what I was talking about but couldn't carry out any calculations.
I once heard an anecdote about Arnol'd that he came to France and lambasted the French mathematicians on a similar basis -- for all their writing, they couldn't carry out "simple" calculations with a clear vision (as once can when one has a physics motivation). It might have been the anecdote was referring to the Serre-Arnold debate -- thanks for that link.
Of course, Arnol'd has... high standards, to put it mildly. In his book, "Problems for children from 5 to 15", he writes in the preface,
"My long experience has shown that, very frequently, dimwits falling at
school behind solve them better than A-grade pupils, since – for their survival
at the back of the classroom – they must permanently think more than required
“for governing the whole Seville and Granada”, as Figaro used to say about
himself, while A-graders cannot catch “what should be multiplied by what”
in these problems. I have also noticed that five year old kids solve similar
problems better than pupils spoiled by coaching, which in their turn cope with
the questions better than university students used to swotting who anyway
beat their professors (the worst in solving these simple problems are Nobel
and Fields prize winners)."
One of the problems is to sum 1/n^2 from 1 to infty (this is for children not older than 15, remember). Not only would I be unable to do this without modern technology (like Fourier analysis), I also find it amazing that Arnold writes:
"prove that the sum is pi^2/6, that is, approximately 3/2",
as though the approximation were harder (or perhaps more important) than finding the exact value. To me, that small comment really underlines just how hands-on he was.
That's interesting. I'd seen Arnold's work on teaching mathematics, and I'm very much aware of the dry formalism in Bourbaki books. However, I was not aware of the Arnold-Serre debate.
But the thing that really caught my eye here is your claim that there were visible effects of Bourbakism polluting mathematics.
Do you have any examples or specifics of that "visible effect". My question is a genuine one, not a challenge to what you are saying.
If you speak/read French, try to have a look at elementary/middle school math books from the 60's/70's in France. It was a disaster and forced a lot of kids to imagine they were bad at maths.
I had discussion about it with Cedric Villani once and he told me that was not a disaster because of the books or the material, but the teachers were not good enough.
Eg. when teaching set theory, teachers were debating what should be a perfect representation of a set (is it an ellipse or circle... etc.?)
Children were completely lost because of this.
If you don’t draw a perfect ellipse to represent a set then you had a bad mark and so on and so forth.
Sure but what good is an education method if the teachers don't understand it. I don't doubt that if you had a great teacher that would work, but considering how focused French education is on mathematics, this was one of the main reason why the social ladder broke.
> I understand, but on the same period, France grew a whole generation of brilliant mathematicians. I don't think that the two are completely unrelated.
If you could prove me that there were more brilliant mathematicians educated in France during the 60's/70's compared to other periods, I'd be interested.
Even if this is the case (which I'm not sure), at least it's pretty clear it lowered the median (whether this is good or not is another debate ;)
But how did Bourbaki influence elementary/middle school mathematics? Bourbaki mathematics is basically at the level of graduate school. Except in countries where a graduate degree is required to teach mathematics, I can't imagine how there would even be a connection between the two worlds. Is that the case in France?
Again, I'm not disputing what you are saying either, merely trying to understand what happened.
Is it still going on in French schools today? Or was it 60's/70's only?
AFAIK the idea at that time was to try to apply the Bourbaky method to teach kids elementary maths - e.g. try to explain the concept of sets before kids know how to add 2 numbers (I'm caricaturing but not that much - you had to explain to kids Peano before addition..). Someone else might have a better answer for that as I'm too young to know. Wikipedia French page on 'New Math' mentions it but not in details.
> Is it still going on in French schools today? Or was it 60's/70's only?
Unless you are in a reaally elitist high school in Paris with a really old teacher, this should'nt happen nowadays.
I'm Czech and this revolutionary method of teaching came here in the 90's, when I entered my grammar school. Probably, it had a political “east vs. west” motivation among others, shortly after the velvet revolution...
The first thing I learned in the 1st year of schools were basic concepts of set theory. We were drawing circles, ellipses and Venn diagrams (even though we didn't call them like that) filled with images of apples, plums and cherries. Teachers explained to us what an intersection, union and set differences are and we were supposed to draw items into one set, but not in another set, etc.
I recall these exercises as funny and playful. They were similar to IQ tests in the sense that the exercise is logical, slightly entertaining, but highly abstract and loosely related to the world you know.
And I think this was the main issue. The 2nd topic we learned was simple arithmetics as in standard educational systems. However, at that time, I didn't see any relationship with the concepts of set theory.
Was the system any good? Hard to say. AFAIK, it was dropped after a few years. Eventually, I obtained a PhD in computer science, so at least, the system wasn't a complete disaster for me. :-)
I think teaching sets like this totally misses the point. Finite sets are trivial (and pretty useless), so there is no use for schoolkids, it is just a waste of time.
On the other hand, arithmetic for very young kids like second-graders is probably also a waste of time. If you waited till they're older (or they express a genuine interest) I expect they'd learn it quickly enough and with less chance of learning mainly to hate math.
Perhaps it made sense in ye olde days when many students would not go on to junior high.
You mean adding and substracting numbers? Personally I think it should be taught (and is here) to first graders. Also here no kid hates math before 5th grade (overly generalizing).
My sample number might be too small (so it is probably anecdotal evidence at most), but I would say german primary schools (which is grade 1-4). To me it looks like the math hate is coming later starting somewhere in secondary school between grade 5 to 7.
Thanks. I based my 'probably' on a few things -- it's admittedly not strong:
- There was an American school about a lifetime ago that tried that strategy, and the principal claimed it worked fine.
- A general impression that before mass education it wasn't uncommon for people who got schooling to start years older than our start, with no impression that they learned arithmetic any worse. E.g. http://www.scientiasocialis.lt/pec/files/pdf/vol57/90-101.Pi... "In general the pupils began their arithmetical instruction at 10–11 and this
education prosecuted for two years."
- Piaget's picture of stages of development (in my vague understanding) suggests that arithmetic beyond a very concrete level would be developmentally unnatural for younger kids, and more natural later. Apparently Montessori schools do better on this score?
- Unschoolers sometimes reach adulthood with less understanding of math than state-schoolers, but if the average is worse, I haven't heard of it. Anecdotally they're fine.
- Hate and ignorance of math is very widespread (I've read similar claims about average French people with their substantially different school system)
- This jibes with my general experience, having gone to school, etc.
- There's not much reason to expect a claim this far from mainstream to have been carefully studied. Maybe it has been and refuted -- I just don't very much respect the status quo and so I expect there are improvements that would 'easy' except for the obstacle that it's very hard to meaningfully change the system. And this strikes me as a plausible (though unambitious) one.
When I learned set theory and relations for the first time, I was in love with the abstractiveness of mathematics. It was so intuitive and I truly felt that I've got the grasp of fundamental principles of maths. The sad thing was, we were introduced to set theory in Grade 11(17 years old), that's too late IMO, it should be taught around Grade 4.
In most countries the schoolbooks had been changed to teaching about sets and stuff in the first grade. Generally more abstract or top-down approach. Old school approach was more bottom-up.
In my country I have been still learning from old schoobooks while classes below had new schoolbooks - thats why I remember that well.
Recently I have been helping 11 year old with classes (remote learning now). And my impression is that this top-down approach is still present but the schoolbook was full of practical life examples (money issues, understanding newspaper articles with pecentages and percent points etc.)
Somehow from the link it seems Arnold didnt manage to make his point in the Arnold Serra debate, and audience thought he is ranting. i find this quite disappointing, since I think he really has a point.
Personal experience:
I came to like mathematics by way of computer science: so first got an understanding and appreciation for discrete maths, logic, etc, and only later on analysis and “phyisics” related mathematics.
It's hard to say exactly why, but I'd wager it's the following: discrete maths is easy to formalize and axiomatize for first year student, so you quickly get a grasp of the “rules of the game”: what axioms you're allowed to use, how an argument works, etc.
On the contrary, when you learn first year analysis, you see all those theorems talking about this (intuitively very simple but) quite complex object – the reals – of which you don't know the axioms (say linearly ordered field something something).
Thus, when you learn about the basics results, you're faced with a kind of two-faced problem: on the one hand, all the basic results are intuitively obvious, while on the other, you have no idea how to build a proof because you don't know what facts you can safely use.
I don't really know how to phrase all that, but discarding the CS/axiomatic side of maths while praising the intuitive physics-inspired one is not the right approach imo.
The Bourbaki volumes are fantastic references for experts, and as with most reference books they are a terrible from an introductory/teaching perspective.
I’ve always viewed Bourbaki as something like the CRC Handbook for pure mathematics: an exceptional reference and a necessary collation of wide-ranging ideas and information, but dry by design, and a poor way for self-learners to understand any new topics.
I also think it’s going a little far to say “remove the beauty” - but it is true that if you don’t already have an appreciation for the specific subject, that beauty is awfully hard to find.
I used Bourbaki as a second text in an advanced math course I took when I was at UC Berkeley in the early 1960's. I remember with delight the spare clarity of the presentation of the material and organic, intuitive organization of the topic. French has always seemed to me to be the language of mathematics, reason, logic (and love).
> French has always seemed to me to be the language of mathematics, reason, logic
For some reason I often find papers and thesis I read in French way more interesting, well thought and presented than most things I read in English. I have a few hypotheses on why I get this impression:
1) the publish or perish culture originated from the Anglo-Saxon world. Until recently the research written in the national language was kinda shielded from it. In Japan for instance, where the higher education is modeled upon the US one, master students are expected to publish at least a conference paper. In France students generally don’t publish anything.
2) the PhD thesis are written differently. Where I’m studying it’s basically slapping three papers together with an introduction and a conclusion. This sometimes leads to awkward thesis and shallow work. Thesis (and HDR) written in Europe are more like a very well structured monograph.
3) the way to use the language is different as well. In French, any intellectual written work will use long and complex sentences (sometimes to a fault) that are cramming few ideas and their relationships. In English the style is to write short sentences, with at most one idea each. I sometimes feel I have to dumb down my writing and splitting sentences while in my native language a Proust-like sentence would be more appropriate.
> For some reason I often find papers and thesis I read in French way more interesting, well thought and presented than most things I read in English.
The much simpler explanation is that there are way more English as a foreign language PhD students than there are French as a foreign language PhD students.
I’m note sure why you bring foreign-speakers doing their PhD in French (the one who does have outstanding language abilities btw), I’m speaking about research output in general. The few thesis I looked from my immediate research entourage are quite weak compared to the few I’ve read from my home country. There is definitely something different in the writing and the expectations between Europe and the New World. To be clear, I don’t think it’s a language issue per se, but a result of the difference in requirements to graduate. For research papers, quality went down because of the rat course.
I bring it up because having non-native speakers being a much larger percentage of the output means a huge hit to the average quality of writing.
I went to a top CS research school and there were several students from China that had such a high language barrier that their papers had to be professionally rewritten by a service the university offered. Their research was Amazing but motivating the problem, describing the methodology, etc all in English just devastated the signal.
So yeah, you’re going to see good quality French writing because French is no longer the lingua Franca of science so a relatively tiny minority of non-French researchers are going to use it.
I wish more science, and generally more work was done this way.
Dispense with ego-flattering prizes, narratives of discovery that put lone geniuses in the spotlight, turning PIs heroes and dictators at the expense of their team...
I understand this is not practical for all fields, but it would alleviate much of the pettiness and unethical behaviors that have become so normalized in academia.
I am no expert but it is not my impression that adopting a group pseudonym eliminated politics or pettiness from the behavior of people inside or outside. But maybe makes the politics a bit less public/transparent?
One criticism I have heard from professional mathematicians is that Bourbaki wasted a whole generation of French talent, pulling the best young mathematicians away from their own useful and interesting research to assign to them a project of marginal benefit if any, because being a member of Bourbaki was in itself prestigious. The work tries to be entirely self-contained instead of part of a conversation, which leads to insularity, arbitrary “not-invented-here” reformulation of established concepts, ignoring outside developments, and lack of historical links and attributions. It is essentially all reworking of previously known mathematics, rather than any new discovery.
I am not a mathematician, and I don’t really have insight into the opportunity cost of research potential for the people involved, but personally I think the Bourbakist style has been very harmful to mathematics: it is entirely dry and formal, eschewing motivation, examples, or pictures. As a reference for professionals it might be okay, but the same style has infected broad swaths of mathematics teaching, and it serves to chase away many newcomers, almost like a kind of hazing ritual.
But Bourbaki is to say the least a controversial group. I don’t think it should uncritically be taken as a model for other researchers.
We need formalism, and we need motivation, examples, and pictures. One isn't better than the other. In particular, formalism isn't about hazing (though I think some people abuse it this way).
Books that claim to appeal more to intuition or rely on visual arguments certainly help a lot with motivating ideas and establishing context, but at a certain point we need to be clear about what exactly we are talking about.
When learning a new topic in math, I personally prefer to start with a more formal, terse text. For me, the texts that focus too much on examples and motivation tend to be chattier -- I have to read an entire paragraph to understand what it is they are getting at, making more challenging the process of chunking the information into digestible bits that I can hold in mind as I shower or go for a walk (and generally less fruitful). Compare to say working through Rudin or Kolmogorov, where I can read a distilled sentence where each word is carefully chosen that I can easily recall and munch on. Part of it is that I have ADHD -- texts that are more formal and less chatty make it easier for me to focus.
That said, I do think Courant had a point with
> Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is unsuitable
for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isolation and atrophy of mathematics. It seems extremely important that students
and instructors should be protected from smug purism.
Basically, math doesn't exist in a vacuum, and I think part of the reason there are so many people with low emotional intelligence in math is this belief that it can. But that's another story
>When learning a new topic in math, I personally prefer to start with a more formal, terse text. For me, the texts that focus too much on examples and motivation tend to be chattier
Soviet books (MIR Publishers) are notoriously terse and to the point. This is what we used in college, and it has made it difficult for me to read a different style, because the content tries my patience.
For example, I couldn't consume Andrew Ng's Coursera ML course, but I appreciated the CS229 recorded at Stanford because he dove directly into the maths part. I didn't want to see slides, I needed to see problem statement, pause the video, work through gradient descent, play the video, and check I got it right. Side note: doing this [pause, working on it, play] has its advantages, it helped avoid a confusion the instructor had in the course with notation, for example, which you had to be sorted out in the later part of the course.
As you said, it is a spectrum of content with different styles for different needs and people, but even for the same person, one might need a style at a certain point and the other at another point. I remember in my third year I used some MIT OCW resources because I felt my brain was shot and I needed to be spoon fed on a topic I was so far behind on.
This is really a good question. On the one hand the work of bourbaki was important, the foundation of math was shaky (I heard there were proof of some theorems and the negative of the same theorem). But then it somehow seems to destroy the spirit of mathematics. Arnold wrote about it:
https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html#:....
To add: I dont know what the solution to this is. So lets say there are two types of approaching math. Approach one is formal/rigid (caring about Axiom of Choice, Set theoretical foundations ..:-)) the other one still (partially) rigorous but more classical in spirit. A big question for me is, how is actually research done? Is it more the first or more the second approach?
So what are possible solutions? I dont know. If you are trained in the first approach, you usually loose the ability to gloss over the gaps in the second approach. If you are trained in the second you cannot (or very difficult) come around the first one.
And then there is physics which uses the sloppy (and often not correct) version of the second approach. I once asked Serge Lang about this (phrasing:'what about physicist getting the right result with wrong mathematics' (thinking about renormalization)). Lang replied:'this is God's way of calculation'.
Oh wow. So it looks like that the reason why I suddenly got 'bad' at math towards the end of high school might be due to the switch to this hyper-formal way of teaching mathematics, instead of using a more geometric/physic way ?
Could be. Although high school (at least where I am) only teaches a tiny drop of mathematics (so the other style is also difficult.). But generally the is an adaptation problem/challenge in the first two semesters of math students where usually only solving a lot of problem and getting them corrected helps.
But dont worry, these things can be overcome (do not fight them at your current stage if you are still a student). Just dont believe the 'hours needed' in the description of the modules (they are politically decided). Generously work around the clock for the first few years.....
edit: more seriously: although working hard is important: go to the office hours of your TA and/or prof and discuss the work you are doing. Usually no student does that and you will learn a lot.
You have to put back the creation of the group in context though.
Back then philosophers/mathematicians were kinda popular (think Sartre, Beauvoir and so on) so making such a group draw a lot of more public attention than it would now.
Sadly, I doubt that such a group could be created now.
In my university in Spain, math students have the Saint Bourbaki day. That day is recognized holiday by the university and they don't have any classes.
I learned about this from the excellent BBC Radio series a brief history of mathematics with Marcus Du Sautoy. Here is the relevant episode: https://www.bbc.co.uk/programmes/b00stcgv
I asked pg directly whether he was nickb, and he said no. More specifically, he said “not only was I not nickb, but I have no idea who he was.”
pg is many things, and he has bent the truth occasionally, but a liar he isn’t. Make of that whatever you will; it’s the closest you’ll get to an answer.
However, the early secret sauce of HN was to hunt for stories manually and to keep the front page interesting every day. (It’s today’s secret sauce too.) One person can do it, but they’d have to spend several hours per day, like a full time job.
This band has adopted the character of Nicolas Bourbaki in their concept album Trench. It's interesting since they commonly use the symbol Ø in their branding, which was introduced in mathematics by Nicolas Bourbaki to denote an empty set.
I also checked to see if they were mentioned! It’s interesting how a band can make you aware of something very obscure (outside of my domain, that is).
I studied algebra and number theory. Bourbaki was unreadable for me. I would compare it to extremely dessicated, tough jerky made out of the oldest bison ever.
I remember reading a set of books by Bourbaki, I distinctly remember being awed by the style and approach. I though this guy was awesome. I find it hard to comprehend that 'he' was actually many authors. Really surprised to find the style matched up.
In my experience I’d say just open one, try to read it, and move on to something else if you can’t decipher it. Eventually one will suit you, or none of them if you’re like me :p
I'd heard of it as well - there was a novel that had a group of aging Biologist get together and publish anonymously under the name.
The clever part of the novel - and I can't even remember the name - was that the "discovery" they made was PCR. It made it seem a lot better then if the novelist had just created some magic invention for the characters. Made it much more grounded - as I remember it. That said, how good could it be if I can't remember the name of the book.
The band Twenty One Pilots references Bourbaki as an integral part of their most recent album, "Trench". They created some pretty intricate puzzles for their fans leading up to the release of the album and it was fun to watch all these (mostly) young people work it out and learn more about math history at the same time.
A group of contemporary Italian intellectuals set up “Nicoletta Bourbaki”, which is “an inquiry group devoted to the merry debunking of right-wing historical revisionism, history-related fake news and neofascist ideologies. [...] The group comprises historians, researchers in various disciplines, writers and activists. The collective pseudonym is a transgender tribute to the ensemble of mathematicians collectively known as «Nicolas Bourbaki»” [0]. The group published quite a bit of material about the manipulation of Wikipedia entries related to WWI and the Italian Eastern border, and other false historical memes being peddled online.
Carl Djerassi was of course himself a world-class drug researcher: he has a good claim to being the most important figure in the development of oral contraceptives for women.
